Nonlinear large deviations

We present a general technique for computing large deviations of nonlinear functions of independent Bernoulli random variables. The method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs. Previous technology, based on Szemerédi's regularity lemma, works only for dense graphs. Applications are also made to exponential random graphs and three-term arithmetic progressions in random sets of integers.